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2008 AMC 10A problems

All 25 problems from the 2008 AMC 10A, with answer choices, worked solutions and hints. Problems are roughly ordered by difficulty: 1–10 are the most approachable, 19–25 the hardest.

Problems

  1. 1Problem 1A bakery owner turns on his doughnut machine at 8:30 am. At 11:10 am the machine has completed one third of the day’s job. At what time will the…Algebra
  2. 2Problem 2A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is 2:1. The ratio of the rectangle’s length to…Algebra
  3. 3Problem 3For the positive integer n, let langle n rangle denote the sum of all the positive divisors of n with the exception of n itself. For example, langle…Algebra
  4. 4Problem 4Suppose that 2/3 of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much as 1/2 of 5 bananas?Algebra
  5. 5Problem 5Which of the following is equal to the product 8/4 · 12/8 · 16/12 … 4n+4/4n … 2008/2004?Algebra
  6. 6Problem 6A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate…Algebra
  7. 7Problem 7The fraction dfrac (3^2008)^2 - (3^2006)^2(3^2007)^2 - (3^2005)^2 simplifies to which of the following?Algebra
  8. 8Problem 8Heather compares the price of a new computer at two different stores. Store A offers 15% off the sticker price followed by a $90 rebate, and store B…Algebra
  9. 9Problem 9Suppose that 2x/3 - x/6 is an integer. Which of the following statements must be true about x?Number Theory
  10. 10Problem 10Each of the sides of a square S_1 with area 16 is bisected, and a smaller square S_2 is constructed using the bisection points as vertices. The same…Geometry
  11. 11Problem 11While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The…Algebra
  12. 12Problem 12In a collection of red, blue, and green marbles, there are 25% more red marbles than blue marbles, and there are 60% more green marbles than red…Algebra
  13. 13Problem 13Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Doug and Dave paint the room together and take a one-hour break for lunch…Algebra
  14. 14Problem 14Older television screens have an aspect ratio of 4:3. That is, the ratio of the width to the height is 4:3. The aspect ratio of many movies is not…Geometry
  15. 15Problem 15Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average…Algebra
  16. 16Problem 16Points A and B lie on a circle centered at O, and ∠ AOB = 60°. A second circle is internally tangent to the first and tangent to both OA and OB. What…Geometry
  17. 17Problem 17An equilateral triangle has side length 6. What is the area of the region containing all points that are outside the triangle and not more than 3…Geometry
  18. 18Problem 18A right triangle has perimeter 32 and area 20. What is the length of its hypotenuse?Algebra
  19. 19Problem 19Rectangle PQRS lies in a plane with PQ = RS = 2 and QR = SP = 6. The rectangle is rotated 90° clockwise about R, then rotated 90° clockwise about the…Geometry
  20. 20Problem 20Trapezoid ABCD has bases AB and CD and diagonals intersecting at K. Suppose that AB = 9, DC = 12, and the area of △ AKD is 24. What is the area of…Geometry
  21. 21Problem 21A cube with side length 1 is sliced by a plane that passes through two diagonally opposite vertices A and C and the midpoints B and D of two opposite…Geometry
  22. 22Problem 22Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he…Counting & Probability
  23. 23Problem 23Two subsets of the set S = {a, b, c, d, e} are to be chosen so that their union is S and their intersection contains exactly two elements. In how…Counting & Probability
  24. 24Problem 24Let k = 2008^2 + 2^2008. What is the units digit of k^2 + 2^k?Number Theory
  25. 25Problem 25A round table has radius 4. Six rectangular place mats are placed on the table. Each place mat has width 1 and length x as shown. They are positioned…Geometry

Practise the same ideas across every year on the topic pages, or browse the full AMC 10 archive.